Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

216 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsG. FEASIBILITY OF A NONANALOG GAMESufficient conditions of the feasibility of an analog game were derived in Section 5.LB.Under these conditions,the expected number of particles present in the domain of simulationtends to zero with a probability 1 if the number of collisions in the history tends to infinity.The conditions concern the norm of the analog transport kernel L(P,F') in Equation (5.14)and are formulated in Theorems 5.1 and 5.2. Most naturally, if the nonanalog transportkernel defined asL(P,P")satisfies the same conditions, then the number of nonanalog particles will also tend to zeroas the simulation goes on and consequently Equation (5.192) will also have a unique boundedsolution. In other words, under the staled conditions, the expected number of collisions inthe nonanalog game without Russian roulette will also be finite with a probability 1. Therefore,it might seem reasonable to impose the conditions of Theorem 5.1 on the nonanalogtransport kernel L as well in order to ensure the feasibility of the nonanalog game. Thereis, however, a little problem with this reasoning; namely, nonanalog games not conformingwith these conditions are widely used and are found feasible. The resolution of this apparentcontradiction is that it is not the number of collisions but the total statistical weight of theparticles present in the system that determines the feasibility. This is so because particleswith low weights can be eliminated from the system with high probability by the applicationof Russian roulette and this elimination will leave the estimation unbiased. In the analoggame, the statistical weights of the particles do not change and therefore the number ofparticles in the system is equivalent to the total weight of them. A finite number of collisionsper history then implies the elimination of all the particles, i.e.. the total weight of theparticles. In a nonanalog game, however, the number of particles has no direct connectionto their total weight.In order to illustrate the situation, let us consider a simple example. Let the quantity tobe estimated be the absorption rate due to a monoenergetic particle migrating in a nonmultiplyinghomogeneous infinite medium. Let the total cross section of the medium be unityand let the scattering probability be c. Accordingly, the reaction rate that is estimated readsi.e., the weighting function isf(P) = 1 - cThis weighting function will serve as the estimator scoring at every collision. Assume thatsurvival biasing is used (cf. Chapter 3.11), i.e., that absorption is replaced by weight reduction.Hence the, nonanalog kernels areT(P,P')dP" -T(P,P')dPDdD for P' = (r + Dw), D S= 0andC(P',F') - C(P',P")/c = C 5(w,oj')